Page 42 - Entrophy Analysis in Thermal Engineering Systems

Teaching entropy 33

(1) if the same amount of work produced by the engine from a given
amount of heat is spent to operate the engine backwards, an equal ther-
mal effect will be produced,
(2) there should not be any temperature change, which may not be due to a
volume change.
In Carnot’s view, not necessarily any engine that could be operated in a
reverse direction would produce a maximum work from a given amount
of heat. Rather, both conditions mentioned above need to be fulfilled.
The proof of Carnot corollaries given in Section 3.2.1 does not use the second
criterion for maximum work production. So, the same reasoning applied to
theCarnotenginein Fig. 3.1 may be used for any heat engine that could be
operated in reverse direction with identical mechanical and thermal effects in
both directions. For instance, an ideal Brayton cycle may be operated in a
reverse direction, as a refrigerator, whereby the amount of work, heat input,
and the rejected heat would be the same in both directions.
In the first series of operation, an amount of heat Q H is supplied to an ideal
Brayton engine to produce a network of W net . The amount of heat rejected by
the cycle is denoted by Q L . In the reverse operation, by providing the same
quantity of rejected heat Q L and spending the same amount of work produced
in the first series of operation, the quantity of heat rejected from the Brayton
refrigerator would be identical to the quantity of heat supplied to the cycle in
the first series of operation. So, one may apply the same reasoning of the proof
of Carnot’s corollaries to show that an ideal Brayton cycle is more efficient
than any engine such as engine A in Fig. 3.1. However, as we know, the effi-
ciency of ideal Brayton cycle is less than that of a Carnot cycle operating
between the same fixed-temperature thermal reservoirs.

3.2.3 Carnot efficiency

The method of derivation of the Carnot efficiency presented in thermody-
namics textbooks relies on Carnot’s second principle, which is used to rea-
son that the efficiency of the Carnot cycle depends solely on the
temperatures of the thermal reservoirs. Although the method of demonstra-
tion of the corollaries suffers from certain issues, we now show that even the
way the Carnot efficiency is derived is not entirely satisfactory.
Based on Carnot’s second corollary, the efficiency of a Carnot cycle is
reasoned to be a function of the reservoirs’ temperatures, thus

Q L
¼ fT H , T L Þ (3.4)
ð
Q H